Finite Group Databasebeta

← #27174411

Subgroups — #27174411

Order 12928.

840 conjugacy classes of subgroups; only the 177 normal subgroups are shown.

normal subgroups only

This group has 840 conjugacy classes of subgroups — too many to draw. Only its 177 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×D404C211yesmaximal, characteristic, complemented
2C2×C4×C22⋊D101C214yesmaximal, complemented
3#20265113C212yesmaximal, complemented
4#20265112C212yesmaximal, complemented
5#20265086C212yesmaximal, complemented
6#20265085C212yesmaximal, complemented
7#20265023C211yesmaximal, characteristic, complemented
8C2×C4×D4×C101C211yesmaximal, characteristic, complemented
9C22×D404C2211yescharacteristic
10C22×C22⋊D101C2214yes—
11C22×C4×D101C2212yes—
12C4×D404—24no—
13C4×C22⋊D101—216no—
14C2×C4⋊Dic101C2211yescharacteristic
15C2×#13108053C2214yes—
16C2×C22⋊Dic101C2214yes—
17C2×#13108051C2212yes—
18C2×#13108051C2214yes—
19C2×C4×Dic101C2212yes—
20#15692162C2212yes—
21#15692162C2212yes—
22C22×D4×C101C2211yescharacteristic
23C23×C4×C101C2212yes—
24C4×D4×C101—24no—
25C2×C4⋊C4×C101C2211yescharacteristic
26C2×C22⋊C4×C101C2212yes—
27C2×C42×C101C2211yescharacteristic
28C23×D101C2312yes—
29C2×D404—24no—
30C2×D404—22no—
31C2×C22⋊D101—28no—
32C2×C22⋊D101—28no—
33C2×C22⋊D101—28no—
34C22×Dic101C2312yes—
35C22×Dic101C2314yes—
36C2×C4×D101—28no—
37C2×C4×D101—24no—
38C4⋊Dic101—22no—
39#13108053—28no—
40C22⋊Dic101—28no—
41#13108051—24no—
42#13108051—28no—
43C4×Dic101D418yescomplemented
44C24×C101C2312yes—
45C2×D4×C101—24no—
46C2×D4×C101—22no—
47C22×C4×C101—24no—
48C22×C4×C101—28no—
49C22×C4×C101C2312yes—
50C22×C4×C101C2311yescharacteristic
51C22×C4×C101C2311yescharacteristic
52C22×C4×C101C2311yescharacteristic
53C4⋊C4×C101—22no—
54C22⋊C4×C101—24no—
55C42×C101D414yescomplemented
56C23×C101—22no—
57C23×C101—24no—
58C23×C101—22no—
59C23×C101—24no—
60C23×C101#4114yes—
61C23×C101C2411yescharacteristic
62C22×D101—24no—
63C22×D101—22no—
64C22×D101—24no—
65C22×D101—22no—
66C22×D101#4114yes—
67D4×C101—28no—
68C2×C4×C101—22no—
69C2×C4×C101—22no—
70C2×C4×C101—22no—
71C2×C4×C101—24no—
72C2×C4×C101—48no—
73C2×C4×C101C2×D414yes—
74C2×C4×C101C2×D412yes—
75C2×C4×C101C2×D412yes—
76C2×C4×C101C2×D412yes—
77C22⋊D101—232no—
78C2×Dic101—24no—
79C2×Dic101—24no—
80C2×Dic101—22no—
81C2×Dic101—24no—
82C2×Dic101C2×D418yes—
83D404—28no—
84C4×D101—48no—
85C22×C101—28no—
86C22×C101—24no—
87C22×C101—42no—
88C22×C101—44no—
89C22×C101—42no—
90C22×C101C22×D412yes—
91C22×C101C22×D411yescharacteristic
92C22×C101C2×#4112yes—
93C22×C101C2×#4112yes—
94C2×D101—24no—
95C2×D101—28no—
96C2×D101—44no—
97C2×D101—42no—
98C2×D101—42no—
99C4×C101—22no—
100C4×C101—22no—
101C4×C101—42no—
102C4×C101#12714yes—
103Dic101—28no—
104Dic101—42no—
105C2×C101—44no—
106C2×C101#52511yescharacteristic
107C2×C101#52514yes—
108C2×C101#52511yescharacteristic
109C2×C101C2×#12711yescharacteristic
110D101—44no—
111#2397—1011nomaximal, Sylow
112C101#239711yescharacteristic, complemented, Sylow
113C2×C4×D4—1014no—
114C2×C4×D4—1011no—
115C2×C4×D4D10111yescharacteristic, complemented
116#399—1012no—
117#399—1012no—
118#392—1012no—
119#392—1012no—
120#376—1011no—
121C22×D4—1014no—
122C22×D4—1011no—
123C22×D4C2×D10111yescharacteristic
124C23×C4—1012no—
125C23×C4C2×D10112yes—
126C4×D4—24no—
127C4×D4—20216no—
128C4×D4—2024no—
129C2×C4⋊C4—1014no—
130C2×C4⋊C4—1011no—
131C2×C4⋊C4C2×D10111yescharacteristic
132C2×C22⋊C4—1014no—
133C2×C22⋊C4—1014no—
134C2×C22⋊C4—1012no—
135C2×C22⋊C4C2×D10112yes—
136C2×C42—1012no—
137C2×C42C2×D10111yescharacteristic
138#95—1012no—
139#95—1012no—
140C24—1012no—
141C24C22×D10112yes—
142C2×D4—24no—
143C2×D4—22no—
144C2×D4—2024no—
145C2×D4—2028no—
146C2×D4—2028no—
147C2×D4—2028no—
148C2×D4—2022no—
149C22×C4—28no—
150C22×C4—24no—
151C22×C4—1014no—
152C22×C4—1012no—
153C22×C4—2028no—
154C22×C4—2024no—
155C22×C4C22×D10112yes—
156C22×C4C22×D10111yescharacteristic
157C22×C4C22×D10111yescharacteristic
158C22×C4C22×D10111yescharacteristic
159C4⋊C4—22no—
160C4⋊C4—2028no—
161C4⋊C4—2022no—
162C22⋊C4—24no—
163C22⋊C4—2028no—
164C22⋊C4—2028no—
165C22⋊C4—2024no—
166C42—1018no—
167C42C22⋊D10114yescomplemented
168C23—24no—
169C23—22no—
170C23—24no—
171C23—22no—
172C23—1014no—
173C23—2024no—
174C23—2022no—
175C23—2022no—
176C23—2024no—
177C23#1310807414yes—
178C23C23×D10111yescharacteristic
179D4—28no—
180D4—2028no—
181D4—20232no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—24no—
185C2×C4—22no—
186C2×C4—48no—
187C2×C4—1018no—
188C2×C4—4048no—
189C2×C4—2024no—
190C2×C4—2022no—
191C2×C4—2024no—
192C2×C4—2024no—
193C2×C4D4×D10114yes—
194C2×C4C2×C22⋊D10112yes—
195C2×C4C2×C22⋊D10112yes—
196C2×C4C2×C22⋊D10112yes—
197C22—28no—
198C22—24no—
199C22—42no—
200C22—44no—
201C22—42no—
202C22—4044no—
203C22—4042no—
204C22—4042no—
205C22—2024no—
206C22—2028no—
207C22C2×D4×D10112yes—
208C22C22×C22⋊D10111yescharacteristic
209C22#41×D10112yes—
210C22C2×#1310807412yes—
211C4—22no—
212C4—22no—
213C4—42no—
214C4—4042no—
215C4—2028no—
216C4#1569225114yes—
217C2—44no—
218C2—4044no—
219C2#2026568411yescharacteristic
220C2#2026568614yes—
221C2#2026568811yescharacteristic
222C2C2×#1569225111yescharacteristic
subgroupG/H
centerC22×C4C22×D101
commutator, socle, cosocleC23×C101C24
FrattiniC23C23×D101
FittingC2×C4×D4×C101C2
hypercenterC2×C4×D4D101
hypocenterC101#2397
LCSG ▷ C23×C101 ▷ C101
UCSC1 ◁ C22×C4 ◁ C2×C4×D4

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×C4×D404C2semidirect
2. C2×C4×C22⋊D101C2semidirect
3. #20265113C2semidirect
4. #20265112C2semidirect
5. #20265086C2semidirect
6. #20265085C2semidirect
7. #20265023C2semidirect
8. C2×C4×D4×C101C2semidirect
43. C4×Dic101D4semidirect
55. C42×C101D4semidirect
112. C101#2397semidirect
115. C2×C4×D4D101semidirect
167. C42C22⋊D101semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×C4×D4042
C2×C4×C22⋊D1012
#202651132
#202651122
#202650862
#202650852
#202650232
C2×C4×D4×C1012
#2397101